Expected value - Properties

Expected value - Properties - More About Expected Value and...

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More About Expected Value and Variance Page 1 of 5 Expected value, , has a number of interesting properties. These aren’t likely to be used in this [] E X course beyond this lesson, but may come into play in a later statistics course. Properties of E X 1. is a linear operator. By this we mean that things that add inside the [ ] add outside. E X 2. . E kk You could apply a “Duh!” to this one. The expected value for any constant outcome is the constant. It comes directly from the definition whether discrete or continuous. Discrete: 11 1 [ ] () s i n c e 1 . nn n ii i i Ek kPX k PX k    Continuous: () s i n c e 1 . bb b aa a k fXd x k fXd x x  Note that this means that practically every property of integration has an equivalent property in expected value. 3. (where f and g are sets of outcomes withe same probability [ ] [ ] E fg E f E g  distribution.) Note that . The last two terms are 1 [ ] ( ) ( ) ( ) n i i E P X f P X g P X the respective expected values. The continuous model property is similar. 4. Combining the two properties give a linear combination: . 12 1 2 [ ] [ ] Ekf kg kEf kEg  Right now you might give a big yawn and a “who cares?” However, we will use these properties in a moment after we define variance. Variance, 2 Var X Variance is defined as follows for the respective discrete or continuous models: Var X Discrete:    2 2 1 n i Var X E X X P X     Continuous:  22 b a Var X E X x f x dx is a measure of spread (dispersion) of data in the probability model. Var X It is the weighted total of all squares of distances from the mean μ to every point in the data.
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More About Expected Value and Variance Page 2 of 5 However, mathematicians dislike the idea of “ a measure”. We want “ the measure”. Hence we want the minimum total of distances between some optimum point and every point in the data set. This is where our properties come in. Let’s assume there is some optimum point , b , then discover what it must be. We also assume that there is a random variable X with an expected value .
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This note was uploaded on 06/16/2011 for the course MAT 211 taught by Professor Seal during the Summer '08 term at ASU.

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Expected value - Properties - More About Expected Value and...

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